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Quantified Modal Relevant Logics

  • Author / Creator
    Ferenz, Nicholas
  • Applications of formal logic often require the language of the logics to be sufficiently expressive, capturing notions such as necessity, possibility, subject-predicate sentences, quantified sentences, and identity. To this end, logics employ modal operators, first order quantifiers, and an identity relation. A logic with this kind of expressive power is called a quantified modal logic. Relevant logics are the logics that ensure the conclusions of an argument are relevant to its premises, and are able capable of making many philosophically motivated inferential distinctions. The quantified modal extensions or relevant logics have not received much attention, partially due to the historical difficulties in developing less expressive relevant logics. This work constructs a general framework for constructing quantified modal relevant logics with identity, focusing on formal semantics. The proof systems given are all Hilbert style axiom systems for a range of quantified modal logics extending the basic affixing logic B. Using insights from work on the regular modal relevant logics, as well as a recent general frame semantics for the quantified relevant logic RQ, I construct a general frame relational semantics for a wide range of logics. To this framework, I add an identity relation in a variety of ways, each of which enjoys a different philosophical motivation. Kremer's relevant indiscernibility approach to identity in relevant logics guides both approaches. The first results in a formal semantics that captures Kremer's informal interpretation of identity, the other builds on Kremer's axiom choice by giving it a semantics for which it is sound and complete. In addition, I explore possible applications in modal naïve set theory. Krajíček's open problem of the consistency of his axiomatization of modal set theory in KT is solved using a modal version of Curry's paradox, which can generalize to show the triviality of numerous other axiomatizations.

  • Subjects / Keywords
  • Graduation date
    Spring 2020
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/r3-7dpe-7e52
  • License
    Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.